In [1]} the authors encode elements of the finitely generated group \(G\)  by words over the group generators
and their inverses, which corresponds to the unary encoding of integers.
Another, more succinct encoding of group elements uses straight-line programs (SLP). These are context-free
grammars that produce a single word. Over a unary alphabet, one can achieve for every word exponential compression with SLPs:
The word \(a^n\)  can be produced by an SLP of size \(O(\log n)\) . This shows that knapsack and subset sum for the group \(\mathbb {Z}\)  with SLP-compressed
group elements correspond to the classical knapsack and subset sum problem with binary encoded numbers.
To distinguish between the two variants, we will speak in this introduction of uncompressed knapsack (resp., subset sum) if the input group
elements are given explicitly by words over the generators. On the other hand, if these words are represented by SLPs, we
will speak of SLP-compressed knapsack (resp., subset sum). Later in this paper, we will only use the uncompressed versions,
and denote these simply with knapsack and subset sum.
